Rubber Bands, Pursuit Games and Shy Couplings

TL;DR

This paper explores geometric properties of domains in Euclidean space, such as rubber bands and loop contractibility, to analyze pursuit-evasion strategies and shy couplings of reflected Brownian motions, establishing conditions for their existence or nonexistence.

Contribution

It introduces geometric notions like rubber bands and loop contractibility to determine the existence of evasion strategies and shy couplings in bounded domains.

Findings

Stable rubber bands imply successful evasion strategies.

Well-contractible loops prevent shy couplings in star-shaped domains.

Reflected Brownian motions in certain domains almost surely come arbitrarily close.

Abstract

In this paper, we consider pursuit-evasion and probabilistic consequences of some geometric notions for bounded and suitably regular domains in Euclidean space that are CAT(kappa) for some kappa > 0. These geometric notions are useful for analyzing the related problems of (a) existence/nonexistence of successful evasion strategies for the Man in Lion and Man problems, and (b) existence/nonexistence of shy couplings for reflected Brownian motions. They involve properties of rubber bands and the extent to which a loop in the domain in question can be deformed to a point without, in between, increasing its loop length. The existence of a stable rubber band will imply the existence of a successful evasion strategy but, if all loops in the domain are well-contractible, then no successful evasion strategy will exist and there can be no co-adapted shy coupling. For example, there can be no shy couplings in bounded and suitably regular star-shaped domains and so, in this setting, any two reflected Brownian motions must almost surely make arbitrarily close encounters as t tends to infinity.